In this paper we investigate numerically the model for pedestrian traffic proposed in [B. Andreianov, C. Donadello, M.D. Rosini, Math. Models Methods Appl. Sci. 24 (2014) 2685−2722]. We prove the convergence of a scheme based on a constraint finite volume method and validate it with an explicit solution obtained in the above reference. We then perform ad hoc simulations to qualitatively validate the model under consideration by proving its ability to reproduce typical phenomena at the bottlenecks, such as Faster Is Slower effect and the Braess’ paradox.
Accepted:
DOI: 10.1051/m2an/2015078
Keywords: Finite volume scheme, scalar conservation law, non-local point constraint, crowd dynamics, capacity drop, Braess’ paradox, Faster Is Slower
@article{M2AN_2016__50_5_1269_0, author = {Andreianov, Boris and Donadello, Carlotta and Razafison, Ulrich and Rosini, Massimiliano D.}, title = {Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1269--1287}, publisher = {EDP-Sciences}, volume = {50}, number = {5}, year = {2016}, doi = {10.1051/m2an/2015078}, zbl = {1370.65042}, mrnumber = {3554543}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015078/} }
TY - JOUR AU - Andreianov, Boris AU - Donadello, Carlotta AU - Razafison, Ulrich AU - Rosini, Massimiliano D. TI - Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1269 EP - 1287 VL - 50 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015078/ DO - 10.1051/m2an/2015078 LA - en ID - M2AN_2016__50_5_1269_0 ER -
%0 Journal Article %A Andreianov, Boris %A Donadello, Carlotta %A Razafison, Ulrich %A Rosini, Massimiliano D. %T Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1269-1287 %V 50 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015078/ %R 10.1051/m2an/2015078 %G en %F M2AN_2016__50_5_1269_0
Andreianov, Boris; Donadello, Carlotta; Razafison, Ulrich; Rosini, Massimiliano D. Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1269-1287. doi : 10.1051/m2an/2015078. http://www.numdam.org/articles/10.1051/m2an/2015078/
New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux. ESAIM Proc. Surv. 50 (2015) 40–65. | DOI | MR | Zbl
,On interface transmission conditions for conservation laws with discontinuous flux of general shape. J. Hyperbolic Differ. Equ. 12 (2015) 343–384. | DOI | MR | Zbl
and ,Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609–645. | DOI | MR | Zbl
, and ,A theory of -dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201 (2011) 27–86. | DOI | MR | Zbl
, and ,Crowd dynamics and conservation laws with nonlocal constraints and capacity drop. Math. Models Methods Appl. Sci. 24 (2014) 2685–2722. | DOI | MR | Zbl
, and ,Riemann problems with non–local point constraints and capacity drop. Math. Biosci. Eng. 12 (2015) 259–278. | DOI | MR | Zbl
, , and ,Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916–938. | DOI | MR | Zbl
and ,A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955–978. | DOI | MR | Zbl
, , and ,Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows. Fire Safety J. 44 (2009) 532–544. | DOI
,Error estimate for Godounov approximation of locally constrained conservation laws SIAM J. Numer. Anal. 50 (2012) 3036–3060. | DOI | MR | Zbl
and ,Numerical Approximation of a Macroscopic Model of Pedestrian Flows. SIAM J. Sci. Comput. 29 (2007) 539–555. | DOI | MR | Zbl
,General constrained conservation laws. Application to pedestrian flow modeling. Netw. Heterog. Media 8 (2013) 433–463. | DOI | MR | Zbl
, and ,Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553–1567. | DOI | MR | Zbl
and ,A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654–675. | DOI | MR | Zbl
and ,Existence of nonclassical solutions in a Pedestrian flow model. Nonlin. Anal. Real World Appl. 10 (2009) 2716–2728. | DOI | MR | Zbl
and ,R.M. Colombo, G. Facchi, G. Maternini and M.D. Rosini, On the continuum modeling of crowds. In vol. 67 of Hyperbolic Problems: Theory, Numerics and Applications, Proc. of Sympos. Appl. Math. AMS, Providence, RI (2009) 517–526. | MR | Zbl
A macroscopic model for pedestrian flows in panic situations. GAKUTO Int. Series Math. Sci. Appl. 32 (2010) 255–272. | MR | Zbl
, , and ,On the modelling and management of traffic. ESAIM: M2AN 45 (2011) 853–872. | DOI | Numdam | MR | Zbl
, and ,E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer Verlag, New York (1996). | MR | Zbl
B.D. Greenshields, A Study of Traffic Capacity, In vol. 14 of Proc. Highway Res. Board (1934) 448–477.
Simulating dynamical features of escape panic. Nature 407 (2000) 487–490. | DOI
, and ,Dynamics of crowd disasters: An empirical study. Phys. Rev. E 75 (2007) 046109. | DOI
, and ,Pedestrian behavior at bottlenecks. Transport. Sci. 39 (2005) 147–159. | DOI
and ,The flow of human crowds. Annu. Rev. Fluid Mech. 35 (2003) 169–182. | DOI | MR | Zbl
,V.A. Kopylow, The study of people’ motion parameters under forced egress situations. Ph.D. thesis, Moscow Civil Engineering Institute (1974).
Experimental study of pedestrian counterflow in a corridor. J. Statist. Mech. 2006 (2006) P10001. | DOI
, , , and ,First order quasilinear equations with several independent variables. Mat. Sb. 81 (1970) 228–255. | MR | Zbl
,R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl
On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads. Proc. Roy. Soc. London Ser. A 229 (1995) 317–345. | MR | Zbl
and ,Microscopic dynamics of pedestrian evacuation. Physica A 354 (2005) 606–618. | DOI
and ,Shock waves on the highway. Oper. Res. 4 (1956) 42–51. | DOI | MR | Zbl
,Nonclassical interactions portrait in a macroscopic pedestrian flow model. J. Differ. Eq. 246 (2009) 408–427. | DOI | MR | Zbl
,M.D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications. Springer, Heidelberg (2013). | MR | Zbl
A. Schadschneider, W. Klingsch, H. Klüpfel and T. Kretz, C. Rogsch and A. Seyfried, Evacuation Dynamics: Empirical Results, Modeling and Applications. In Extreme Environmental Events, edited by R.A. Meyers. Springer (2011) 517–550.
A. Seyfried, T. Rupprecht, A. Winkens, O. Passon, B. Steffen, W. Klingsch and M. Boltes, Capacity Estimation for Emergency Exits and Bottlenecks. In Interflam 2007 (2007) 247–258.
Experimental evidence of the “Faster is Slower” effect in the evacuation of ants. Safety Sci. 50 (2012) 1584–1588. | DOI
, and ,A non-equilibrium traffic model devoid of gas-like behavior. Transport. Res. Part B 36 (2002) 275–290. | DOI
,Empirical study of a unidirectional dense crowd during a real mass event. Physica. A 392 (2013) 2781–2791. | DOI | MR | Zbl
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